Now given these assumption, we are able to show that the constant akin to \(\sigma (\hat{S})\)  appears in the analysis of this algorithm by utilizing the notions from [1]}. We sacrifice generality for the sake of brevity, and assume that range of \((x)\)  spans whole \(^d\)  for each \(x \in \) . Then, the following quantities resembling \(\sigma (\hat{S})\)  appear in the convergence analysis of the update rule (REF )
\(\hat{\sigma }(\hat{S},x) = \lambda _{\min } \left( _{\hat{S}} \left[ ^{1/2}(x)((x)_{\hat{S}})^+ (x)^{1/2} \right] \right)\) 
